One important property of the analytic capacity \(c_B \)  is that it is distance decreasing with respect to any given holomorphic map \(f:X \rightarrow Y\) ,
where \(X, Y\)  are hyperbolic Riemann surfaces, cf. [1]}:
\( f^*\left(c_{B; Y}(z) |dz| \right) \le c_{B; X}(z) |dz|,\) 
